A Comprehensive Stress Test of Truncated Hilbert Space Bases against Green's function Monte Carlo in U(1) Lattice Gauge Theory

TL;DR

The paper tackles the challenge of digitising the infinite‑dimensional link Hilbert space in Hamiltonian lattice gauge theories by comparing several truncation schemes, including electric, magnetic, and a new plaquette‑state functional basis built from single‑plaquette eigenstates. It demonstrates that the plaquette‑state basis yields the most accurate results at a fixed operator dimension in 2D U(1) across a wide range of couplings and remains feasible in 3D, with Green’s function Monte Carlo providing cross‑checks where possible. Extending to SU(2) shows promising preliminary results for the plaquette basis, though a reliable local dual formulation for non‑Abelian theories remains an open challenge. The work supports the plaquette‑state approach as a compact, low‑truncation digitisation for tensor‑network and quantum computing simulations of LGTs and highlights GFMC as a valuable benchmarking tool for Hamiltonian methods.

Abstract

A representation of Lattice Gauge Theories (LGT) suitable for simulations with tensor network state methods or with quantum computers requires a truncation of the Hilbert space to a finite dimensional approximation. In particular for U(1) LGTs, several such truncation schemes are known, which we compare with each other using tensor network states. We show that a functional basis obtained from single plaquette Hamiltonians -- which we call plaquette state basis -- outperforms the other schemes in two spatial dimensions for plaquette, ground state energy and mass gap, as it is delivering accurate results for a wide range of coupling strengths with a minimal number of basis states. We also show that this functional basis can be efficiently used in three spatial dimensions. Green's function Monte Carlo appears to be a highly useful tool to verify tensor network states results, which deserves further investigation in the future.

Figures (10)

• Figure 1: Sketch of the $3 \times 3$ system in the original formulation on the left and the dual formulation on the right.
• Figure 2: Spectrum and eigenstates of the single plaquette system.
• Figure 3: The ground state energy (top) and ground state plaquette expectation value (bottom) as a function of the coupling $g^2$ for a $4 \times 4$ system with U$(1)$ gauge. This is done for the original Kogut-Susskind Hamiltonian using electric basis operators (purple), and for the dual formulation using electric (red), magnetic (orange) and plaquette state operators (blue). All truncations are chosen such that $d_{\mathrm{op}} = 5$. Their results are compared to GFMC results shown by the black crosses.
• Figure 4: The mass gap $M$ as a function of the coupling $g^2$ for the same system and operators as \ref{['fig:u14x4ResultsGnd']}.
• Figure 5: The relative deviations of the ground state energy (top) and the plaquette expectation value (bottom) at a coupling of $g^2=0.1$ as a function of the system size $L$. This is done for magnetic operators at three different truncations (orange, red, pink) and for plaquette state operators at two different truncations (blue, navy).